Algebra Examples

$\frac{\sqrt{- 16}}{3 - 3 i + (1 - 2 i)}$

Step 1

Remove parentheses.

$\frac{\sqrt{- 16}}{3 - 3 i + 1 - 2 i}$

Step 2

Pull out imaginary unit $i$ .

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Step 2.1

Rewrite $- 16$ as $- 1 (16)$ .

$\frac{\sqrt{- 1 (16)}}{3 - 3 i + 1 - 2 i}$

Step 2.2

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$\frac{\sqrt{- 1} \cdot \sqrt{16}}{3 - 3 i + 1 - 2 i}$

Step 2.3

Rewrite $\sqrt{- 1}$ as $i$ .

$\frac{i \cdot \sqrt{16}}{3 - 3 i + 1 - 2 i}$

Step 3

Multiply the numerator and denominator of $\frac{i \cdot \sqrt{16}}{4 - 5 \cdot i}$ by the conjugate of $4 - 5 \cdot i$ to make the denominator real.

$\frac{i \cdot \sqrt{16}}{4 - 5 \cdot i} \cdot \frac{4 + 5 i}{4 + 5 i}$

Step 4

Multiply.

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Step 4.1

Combine.

$\frac{i \cdot \sqrt{16} (4 + 5 i)}{(4 - 5 \cdot i) (4 + 5 i)}$

Step 4.2

Simplify the numerator.

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Step 4.2.1

Rewrite $16$ as $4^{2}$ .

$\frac{i \cdot \sqrt{4^{2}} (4 + 5 i)}{(4 - 5 \cdot i) (4 + 5 i)}$

Step 4.2.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{i \cdot 4 (4 + 5 i)}{(4 - 5 \cdot i) (4 + 5 i)}$

Step 4.2.3

Move $4$ to the left of $i$ .

$\frac{4 \cdot i (4 + 5 i)}{(4 - 5 \cdot i) (4 + 5 i)}$

Step 4.2.4

Apply the distributive property.

$\frac{4 i \cdot 4 + 4 i (5 i)}{(4 - 5 \cdot i) (4 + 5 i)}$

Step 4.2.5

Multiply $4$ by $4$ .

$\frac{16 i + 4 i (5 i)}{(4 - 5 \cdot i) (4 + 5 i)}$

Step 4.2.6

Multiply $4 i (5 i)$ .

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Step 4.2.6.1

Multiply $5$ by $4$ .

$\frac{16 i + 20 i i}{(4 - 5 \cdot i) (4 + 5 i)}$

Step 4.2.6.2

Raise $i$ to the power of $1$ .

$\frac{16 i + 20 (i^{1} i)}{(4 - 5 \cdot i) (4 + 5 i)}$

Step 4.2.6.3

Raise $i$ to the power of $1$ .

$\frac{16 i + 20 (i^{1} i^{1})}{(4 - 5 \cdot i) (4 + 5 i)}$

Step 4.2.6.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{16 i + 20 i^{1 + 1}}{(4 - 5 \cdot i) (4 + 5 i)}$

Step 4.2.6.5

Add $1$ and $1$ .

$\frac{16 i + 20 i^{2}}{(4 - 5 \cdot i) (4 + 5 i)}$

Step 4.2.7

Simplify each term.

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Step 4.2.7.1

Rewrite $i^{2}$ as $- 1$ .

$\frac{16 i + 20 \cdot - 1}{(4 - 5 \cdot i) (4 + 5 i)}$

Step 4.2.7.2

Multiply $20$ by $- 1$ .

$\frac{16 i - 20}{(4 - 5 \cdot i) (4 + 5 i)}$

Step 4.2.8

Reorder $16 i$ and $- 20$ .

$\frac{- 20 + 16 i}{(4 - 5 \cdot i) (4 + 5 i)}$

Step 4.3

Simplify the denominator.

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Step 4.3.1

Expand $(4 - 5 \cdot i) (4 + 5 i)$ using the FOIL Method.

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Step 4.3.1.1

Apply the distributive property.

$\frac{- 20 + 16 i}{4 (4 + 5 i) - 5 \cdot i (4 + 5 i)}$

Step 4.3.1.2

Apply the distributive property.

$\frac{- 20 + 16 i}{4 \cdot 4 + 4 (5 i) - 5 \cdot i (4 + 5 i)}$

Step 4.3.1.3

Apply the distributive property.

$\frac{- 20 + 16 i}{4 \cdot 4 + 4 (5 i) - 5 \cdot i \cdot 4 - 5 \cdot i (5 i)}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Multiply $4$ by $4$ .

$\frac{- 20 + 16 i}{16 + 4 (5 i) - 5 \cdot i \cdot 4 - 5 \cdot i (5 i)}$

Step 4.3.2.2

Multiply $5$ by $4$ .

$\frac{- 20 + 16 i}{16 + 20 i - 5 \cdot i \cdot 4 - 5 \cdot i (5 i)}$

Step 4.3.2.3

Multiply $4$ by $- 5$ .

$\frac{- 20 + 16 i}{16 + 20 i - 20 \cdot i - 5 \cdot i (5 i)}$

Step 4.3.2.4

Multiply $5$ by $- 5$ .

$\frac{- 20 + 16 i}{16 + 20 i - 20 \cdot i - 25 \cdot i i}$

Step 4.3.2.5

Raise $i$ to the power of $1$ .

$\frac{- 20 + 16 i}{16 + 20 i - 20 \cdot i - 25 \cdot (i^{1} i)}$

Step 4.3.2.6

Raise $i$ to the power of $1$ .

$\frac{- 20 + 16 i}{16 + 20 i - 20 \cdot i - 25 \cdot (i^{1} i^{1})}$

Step 4.3.2.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{- 20 + 16 i}{16 + 20 i - 20 \cdot i - 25 \cdot i^{1 + 1}}$

Step 4.3.2.8

Add $1$ and $1$ .

$\frac{- 20 + 16 i}{16 + 20 i - 20 \cdot i - 25 \cdot i^{2}}$

Step 4.3.2.9

Subtract $20 \cdot i$ from $20 i$ .

$\frac{- 20 + 16 i}{16 + 0 - 25 \cdot i^{2}}$

Step 4.3.2.10

Add $16$ and $0$ .

$\frac{- 20 + 16 i}{16 - 25 \cdot i^{2}}$

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Rewrite $i^{2}$ as $- 1$ .

$\frac{- 20 + 16 i}{16 - 25 \cdot - 1}$

Step 4.3.3.2

Multiply $- 25$ by $- 1$ .

$\frac{- 20 + 16 i}{16 + 25}$

Step 4.3.4

Add $16$ and $25$ .

$\frac{- 20 + 16 i}{41}$

Step 5

Split the fraction $\frac{- 20 + 16 i}{41}$ into two fractions.

$\frac{- 20}{41} + \frac{16 i}{41}$

Step 6

Move the negative in front of the fraction.

$- \frac{20}{41} + \frac{16 i}{41}$